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Determine the equation of the tangent line and the quadratic approximation to the function z = e2" at the point r = 0.

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter6: Topics In Analytic Geometry

Section6.2: Introduction To Conics: parabolas

Problem 4ECP: Find an equation of the tangent line to the parabola y=3x2 at the point 1,3.

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Question

Determine the equation of the tangent line and the quadratic approximation to the function
z = e2" at the point r = 0.
y A
Use the oiven granhs of f and a to find each derivative:

Expert Solution

Step 1

Given function is: $z = e^{2 x}$

To find the equation of the tangent at a point $x = x_{0}$ to curve $y = f (x)$ , determine the slope of the tangent which is $\frac{d y}{d t}$ at $x = x_{0}$ .

Differentiate the function $z = e^{2 x}$

$\frac{d z}{d t} = 2 e^{2 x}$

The slope at $x = 0$ is:

$\begin{array}{rcl} {(\frac{d z}{d t})}_{x = 0} & = & 2 e^{2 (0)} \\ = & 2 \end{array}$

At $x = 0$ , the value of the function is:

$\begin{array}{rcl} z_{0} & = & e^{2 (0)} \\ = & 1 \end{array}$

Then, the equation of the tangent is given by the formula:

$z - z_{0} = {(\frac{d z}{d x})}_{x_{0}} (x - x_{0}) \Rightarrow z - 1 = 2 (x - 0) \Rightarrow z = 2 x + 1$

Therefore, the equation of the tangent at $x = 0$ is $z = 2 x + 1$

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